Changing Relationship Between Input and Output: Sigmoidal Thinking
Cem Surmen, CFA
Risk and Compliance Director, LeasePlan Hungary / ALD Automotive Hungary
My first encounter with the concept of stochastic thinking was a true moment of enlightenment for me. Handling uncertainty as a distribution of probabilities rather than a mere point estimate was meaningfully inspiring. Life was beyond being full of uncertainties; it was the uncertainty itself and this new-to-me concept was a gate to a totally new way of interpreting it. Not being sure why I was so sentimental about it, similar feelings proliferate inside me towards another way of understanding change and growth; 'Sigmoidal Thinking'. Its varying patterns between accelerating and diminishing reactions to the input is what grabs my attention, I think.
After a short (and maybe boring for many) mathematical introduction, I will go over some of the areas that this beautiful and pleasant little function is applicable in real life.
Mathematical Background
Sigmoidal function is a special type of logarithmic function. Its (simplified) formula and graph look like this:
Basically, it tells, actually it reminds us that the output will not be a linear function of input. At fist stage, the returns will be discouragingly low. After a certain point though, the output will start to accelerate and later, again after one point, it will start to moderate. Sigmoid function can get similar but different shapes. In this slightly more complex version below, we can alter the pace of acceleration and diminishing with changing c1 and c2 coefficients. As c1 approaches to 0, the function becomes more linear.
Sigmoid curve also works as the cumulative distribution and is the integral of normal distribution function. Or put it another way, normal distribution is the derivative of sigmoid function.
But what is the significance of this simple curve? What makes it worthwhile to be a subject to this post on Linkedin? To me, it tells a lot about what to expect to get from my efforts that I set forth. This is what I especially like about it.
A Familiar Curve that was not Named as Sigmoid: Learning Curve
When studying a totally new topic that you are not familiar at, the things that you read will sound like gibberish at first stage. Progress will be very limited. But as you keep repeating and reading, things will start to make sense. After one point as you grasp the essence, learning will accelerate, and you will be able to really understand what the topic is all about. You will build up the things easily and feel that you are on top of what you are studying. Yet, no matter how much you study, most probably there will still remain some points that you cannot comprehend. This is the very point that the slope of the curve start to decrease, and the marginal learning reduces. But at that point you will already be proficient enough to cover most of the subject. And this is where our lovely sigmoidal curve turns into the notorious learning curve.
This is true for sports too. Or playing a musical instrument. Process is the same; first frustration, then fast improvement and finally diminishing returns. Quitters are the ones who cannot make it through the first phase. The ones who are patient and determined enough can reach to the second phase and achieve growth.
The uses of sigmoid function are much more, including machine learning, artificial neural networks, medical, biology, agriculture, architecture, options, engineering, risk management, etc. It is impossible to make an exhaustive list and coverage here but let us take a quick look to some of those topics broadly.
Other Uses; AI, Options, Marketing and many more
In neural networks, sigmoid function is used to determine the passage of data through neuron layers. The transfer of data is not linear. The neuron that functions sigmoidally is often called a ‘sigmoid unit’. The output of the sigmoid unit thus, will not be a linear function of the weighted average of the data received from previous layers. Sigmoid unit acts like a light dimmer to transfer the data. It is used in deep learning and supervised learning algorithms.
In the universe of options, sensitivities of the price of the options to certain underlying parameters are measured by so-called Greeks. Delta (uppercase?Δ, lowercase?δ?or???) is the sensitivity of the option price to the price of the underlying. It is the first derivative of option price with respect to underlying price and hence the first-order determinant of option price.
Because the value of a long call is directly proportional and the value of a long put is inversely proportional to the price of the underlying, call deltas range from 0 to 1 and put deltas range from 0 to ?1. ?When the share price is close to the strike price, a rough approximation is that a long ATM option will have a delta that is approximately 0.5 (for a call) or ?0.5 (for a put). Below graph shows the delta for the put and call versus share price.
Source: CFA Institute
Option prices also change with time. The delta of a long call position, measured at different times to maturities, t, all else being equal, is given below. As the maturity approaches, delta curve gets closer to the pay-off profile of the option. Time decay of option prices (option price sensitivity to remaining time) is measured by another Greek, Theta (uppercase Θ, lowercase θ), which is out of the scope of this reading.
Options trading is a very broad area, which after a certain depth, direct daylight cannot reach and may make Mariana Trench jealous. The profit/loss profile of Bull/Bear Call/Put Spread (one of the most popular option trading strategies) is another example of sigmoid curves. In below chart, there is the profit/loss profile of a hypothetical Bull Put Spread, in which the more expensive option (the one closer to the money) is sold and the option with a strike farther away from the money (the less expensive one) is bought,?just as with a bear call.
Hipkins and Cowie (2016), have put forward certain applications of sigmoid curve to social sciences. Their observations on the use of technology in marketing was worth noting. “Overlapping s-curves are used to describe changes in the way technology is employed in advertising. New technologies begin slowly, develop rapidly and consolidate. At the same time, only early adopters initially use the technology, then more people do so, and ultimately use becomes prevalent. The place where an s-curve and its successor overlap is seen as a critical space of opportunity. Those who anticipate the ‘next’ big thing gain a commercial edge on the market. Importantly they do not wait for comprehensive consolidation of the technology itself and/or extensive diffusion and uptake. Instead, they begin their research and development before the pace of change triggered by the previous innovation begins to slow down and level out.”
Source: Hipkins and Cowie (2016), ‘The Sigmoid Curve as a Metaphor for Growth and Change’ Teachers and Curriculum, Volume 16, Issue 2, 2016
Sigmoid curve is with us at Covid vaccination statistics too. Vaccination rates clearly displays this pattern of slow start, acceleration and slowing down again.
In their article, Yin, Goudriaan, Lantinga, Vos and Spiertz (2003) detected that growth patterns of plants follow a bell-shaped curve, which resembles nothing but a negative-skewed normal distribution. Thus, the total weight of the plant, which could be modeled with the integral of growth function is a sigmoidal curve.
In the context of risk management, the contribution of additional one unit of leverage to an all-equity portfolio is minimal when the proportion of the additional leverage to the total is limited. As the amount of leverage increases, the overall risk of the portfolio starts to accelerate. After one point though, the marginal increase in portfolio risk starts to diminish, while total risk keeps increasing.
Why It Matters?
Apart from all those different areas of use, the reason that I find sigmoid curve so relevant is that it reminds me of perseverance. It is my carrot. It helps me to stay up and continue.
It tells another important lesson too. After reaching the critical level of accumulated output, whatever that level is, if the pace of output starts to lag that of input, let it go. There is not much need to push any further. Spend you resource at somewhere else. Somewhere more productive. Somewhere productive, actually..
Branch Manager @ Garanti BBVA | Banking
2 年“.. it reminds us that the output will not be a linear function of input. “ ????